Problem 8 on pg. 16 of Counting the art of enumerative combinatorics by George E. Martin
- How many ways can we partition mn distinguishable objects into m piles of n objects each.
The answer is $\frac{mn!}{n!^{m}\cdot m!}$
I visualize the problem as a mississippi problem with n of each of m distinguishable letters. This results in
$\frac{mn!}{n!^{m}}$
I don't understand where the additional m! comes from. Is partition in this case a key word that requires a different counting technique? Is it because the piles them selves are indistinguishable? How do I visualize this extra constraint on the problem?